Projectively coresolved Gorenstein flat dimension of groups

Abstract

In this paper, we introduce and study the projectively coresolved Gorenstein flat dimension of a group $G$ over a commutative ring $R$ and we prove that this dimension enjoys all the properties of the cohomological and the Gorenstein cohomological dimension. We also provide good estimations for the Gorenstein global dimension of $RG$ in terms of this dimension and the Gorenstein global dimension of $R$. Moreover, we study special cases of groups, such as $\textsc{\textbf{lh}}\mathfrak{F}$-groups, and show that for such a group every Gorenstein projective $RG$-module is Gorenstein flat when the global dimension of $R$ is finite.